@unpublished{BlanchardDelattreRoquain2012,
author = {Blanchard, Gilles and Delattre, Sylvain and Roquain, {\´E}tienne},
title = {Testing over a continuum of null hypotheses},
url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56877},
year = {2012},
abstract = {We introduce a theoretical framework for performing statistical hypothesis testing simultaneously over a fairly general, possibly uncountably infinite, set of null hypotheses. This extends the standard statistical setting for multiple hypotheses testing, which is restricted to a finite set. This work is motivated by numerous modern applications where the observed signal is modeled by a stochastic process over a continuum. As a measure of type I error, we extend the concept of false discovery rate (FDR) to this setting. The FDR is defined as the average ratio of the measure of two random sets, so that its study presents some challenge and is of some intrinsic mathematical interest. Our main result shows how to use the p-value process to control the FDR at a nominal level, either under arbitrary dependence of p-values, or under the assumption that the finite dimensional distributions of the p-value process have positive correlations of a specific type (weak PRDS). Both cases generalize existing results established in the finite setting, the latter one leading to a less conservative procedure. The interest of this approach is demonstrated in several non-parametric examples: testing the mean/signal in a Gaussian white noise model, testing the intensity of a Poisson process and testing the c.d.f. of i.i.d. random variables. Conceptually, an interesting feature of the setting advocated here is that it focuses directly on the intrinsic hypothesis space associated with a testing model on a random process, without referring to an arbitrary discretization.},
language = {en}
}
@unpublished{BlanchardMathe2012,
author = {Blanchard, Gilles and Math{\´e}, Peter},
title = {Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration},
url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-57117},
year = {2012},
abstract = {The authors discuss the use of the discrepancy principle for statistical inverse problems, when the underlying operator is of trace class. Under this assumption the discrepancy principle is well defined, however a plain use of it may occasionally fail and it will yield sub-optimal rates. Therefore, a modification of the discrepancy is introduced, which takes into account both of the above deficiencies. For a variety of linear regularization schemes as well as for conjugate gradient iteration this modification is shown to yield order optimal a priori error bounds under general smoothness assumptions. A posteriori error control is also possible, however at a sub-optimal rate, in general. This study uses and complements previous results for bounded deterministic noise.},
language = {en}
}
@article{BlanchardKawanabeSugiyamaetal.2006,
author = {Blanchard, Gilles and Kawanabe, Motoaki and Sugiyama, Masashi and Spokoiny, Vladimir G. and M{\"u}ller, Klaus-Robert},
title = {In search of non-Gaussian components of a high-dimensional distribution},
issn = {1532-4435},
year = {2006},
abstract = {Finding non-Gaussian components of high-dimensional data is an important preprocessing step for efficient information processing. This article proposes a new linear method to identify the '' non-Gaussian subspace '' within a very general semi-parametric framework. Our proposed method, called NGCA (non-Gaussian component analysis), is based on a linear operator which, to any arbitrary nonlinear (smooth) function, associates a vector belonging to the low dimensional non-Gaussian target subspace, up to an estimation error. By applying this operator to a family of different nonlinear functions, one obtains a family of different vectors lying in a vicinity of the target space. As a final step, the target space itself is estimated by applying PCA to this family of vectors. We show that this procedure is consistent in the sense that the estimaton error tends to zero at a parametric rate, uniformly over the family, Numerical examples demonstrate the usefulness of our method},
language = {en}
}
@article{KawanabeBlanchardSugiyamaetal.2006,
author = {Kawanabe, Motoaki and Blanchard, Gilles and Sugiyama, Masashi and Spokoiny, Vladimir G. and M{\"u}ller, Klaus-Robert},
title = {A novel dimension reduction procedure for searching non-Gaussian subspaces},
issn = {0302-9743},
doi = {10.1007/11679363_19},
year = {2006},
abstract = {In this article, we consider high-dimensional data which contains a low-dimensional non-Gaussian structure contaminated with Gaussian noise and propose a new linear method to identify the non-Gaussian subspace. Our method NGCA (Non-Gaussian Component Analysis) is based on a very general semi-parametric framework and has a theoretical guarantee that the estimation error of finding the non-Gaussian components tends to zero at a parametric rate. NGCA can be used not only as preprocessing for ICA, but also for extracting and visualizing more general structures like clusters. A numerical study demonstrates the usefulness of our method},
language = {en}
}
@article{BlanchardMathe2012,
author = {Blanchard, Gilles and Mathe, Peter},
title = {Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration},
series = {Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data},
volume = {28},
journal = {Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data},
number = {11},
publisher = {IOP Publ. Ltd.},
address = {Bristol},
issn = {0266-5611 (print)},
doi = {10.1088/0266-5611/28/11/115011},
pages = {23},
year = {2012},
abstract = {The authors discuss the use of the discrepancy principle for statistical inverse problems, when the underlying operator is of trace class. Under this assumption the discrepancy principle is well defined, however a plain use of it may occasionally fail and it will yield sub-optimal rates. Therefore, a modification of the discrepancy is introduced, which corrects both of the above deficiencies. For a variety of linear regularization schemes as well as for conjugate gradient iteration it is shown to yield order optimal a priori error bounds under general smoothness assumptions. A posteriori error control is also possible, however at a sub-optimal rate, in general. This study uses and complements previous results for bounded deterministic noise.},
language = {en}
}
@article{KloftBlanchard2012,
author = {Kloft, Marius and Blanchard, Gilles},
title = {On the Convergence Rate of l(p)-Norm Multiple Kernel Learning},
series = {JOURNAL OF MACHINE LEARNING RESEARCH},
volume = {13},
journal = {JOURNAL OF MACHINE LEARNING RESEARCH},
publisher = {MICROTOME PUBL},
address = {BROOKLINE},
issn = {1532-4435 (print)},
pages = {2465 -- 2502},
year = {2012},
abstract = {We derive an upper bound on the local Rademacher complexity of l(p)-norm multiple kernel learning, which yields a tighter excess risk bound than global approaches. Previous local approaches analyzed the case p - 1 only while our analysis covers all cases 1 <= p <= infinity, assuming the different feature mappings corresponding to the different kernels to be uncorrelated. We also show a lower bound that shows that the bound is tight, and derive consequences regarding excess loss, namely fast convergence rates of the order O( n(-)1+alpha/alpha where alpha is the minimum eigenvalue decay rate of the individual kernels.},
language = {en}
}
@article{BlanchardDelattreRoquain2014,
author = {Blanchard, Gilles and Delattre, Sylvain and Roquain, Etienne},
title = {Testing over a continuum of null hypotheses with False Discovery Rate control},
series = {Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability},
volume = {20},
journal = {Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability},
number = {1},
publisher = {International Statistical Institute},
address = {Voorburg},
issn = {1350-7265 (print)},
doi = {10.3150/12-BEJ488},
pages = {304 -- 333},
year = {2014},
abstract = {We consider statistical hypothesis testing simultaneously over a fairly general, possibly uncountably infinite, set of null hypotheses, under the assumption that a suitable single test (and corresponding p-value) is known for each individual hypothesis. We extend to this setting the notion of false discovery rate (FDR) as a measure of type I error. Our main result studies specific procedures based on the observation of the p-value process. Control of the FDR at a nominal level is ensured either under arbitrary dependence of p-values, or under the assumption that the finite dimensional distributions of the p-value process have positive correlations of a specific type (weak PRDS). Both cases generalize existing results established in the finite setting. Its interest is demonstrated in several non-parametric examples: testing the mean/signal in a Gaussian white noise model, testing the intensity of a Poisson process and testing the c.d.f. of i.i.d. random variables.},
language = {en}
}
@article{BlanchardDickhausRoquainetal.2014,
author = {Blanchard, Gilles and Dickhaus, Thorsten and Roquain, Etienne and Villers, Fanny},
title = {On least favorable configurations for step-up-down tests},
series = {Statistica Sinica},
volume = {24},
journal = {Statistica Sinica},
number = {1},
publisher = {Statistica Sinica, Institute of Statistical Science, Academia Sinica},
address = {Taipei},
issn = {1017-0405 (print)},
doi = {10.5705/ss.2011.205},
pages = {1 -- U31},
year = {2014},
language = {en}
}
@unpublished{BlanchardMuecke2016,
author = {Blanchard, Gilles and M{\"u}cke, Nicole},
title = {Optimal rates for regularization of statistical inverse learning problems},
volume = {5},
number = {5},
publisher = {Universit{\"a}tsverlag Potsdam},
address = {Potsdam},
issn = {2193-6943 (online)},
url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-89782},
pages = {36},
year = {2016},
abstract = {We consider a statistical inverse learning problem, where we observe the image of a function f through a linear operator A at i.i.d. random design points X_i, superposed with an additional noise. The distribution of the design points is unknown and can be very general. We analyze simultaneously the direct (estimation of Af) and the inverse (estimation of f) learning problems. In this general framework, we obtain strong and weak minimax optimal rates of convergence (as the number of observations n grows large) for a large class of spectral regularization methods over regularity classes defined through appropriate source conditions. This improves on or completes previous results obtained in related settings. The optimality of the obtained rates is shown not only in the exponent in n but also in the explicit dependence of the constant factor in the variance of the noise and the radius of the source condition set.},
language = {en}
}
@unpublished{BlanchardKraemer2016,
author = {Blanchard, Gilles and Kr{\"a}mer, Nicole},
title = {Convergence rates of kernel conjugate gradient for random design regression},
volume = {5},
number = {8},
publisher = {Universit{\"a}tsverlag Potsdam},
address = {Potsdam},
issn = {2193-6943 (online)},
url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-94195},
pages = {31},
year = {2016},
abstract = {We prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. Following the setting introduced in earlier related literature, we study so-called "fast convergence rates" depending on the regularity of the target regression function (measured by a source condition in terms of the kernel integral operator) and on the effective dimensionality of the data mapped into the kernel space. We obtain upper bounds, essentially matching known minimax lower bounds, for the L^2 (prediction) norm as well as for the stronger Hilbert norm, if the true regression function belongs to the reproducing kernel Hilbert space. If the latter assumption is not fulfilled, we obtain similar convergence rates for appropriate norms, provided additional unlabeled data are available.},
language = {en}
}
@book{Blanchard2011,
author = {Blanchard, Gilles},
title = {Komplexit{\"a}tsanalyse in Statistik und Lerntheorie : Antrittsvorlesung 2011-05-04},
publisher = {Univ.-Bibl.},
address = {Potsdam},
year = {2011},
abstract = {Gilles Blanchards Vortrag gew{\"a}hrt Einblicke in seine Arbeiten zur Entwicklung und Analyse statistischer Eigenschaften von Lernalgorithmen. In vielen modernen Anwendungen, beispielsweise bei der Schrifterkennung oder dem Spam- Filtering, kann ein Computerprogramm auf der Basis vorgegebener Beispiele automatisch lernen, relevante Vorhersagen f{\"u}r weitere F{\"a}lle zu treffen. Mit der mathematischen Analyse der Eigenschaften solcher Methoden besch{\"a}ftigt sich die Lerntheorie, die mit der Statistik eng zusammenh{\"a}ngt. Dabei spielt der Begriff der Komplexit{\"a}t der erlernten Vorhersageregel eine wichtige Rolle. Ist die Regel zu einfach, wird sie wichtige Einzelheiten ignorieren. Ist sie zu komplex, wird sie die vorgegebenen Beispiele "auswendig" lernen und keine Verallgemeinerungskraft haben. Blanchard wird erl{\"a}utern, wie Mathematische Werkzeuge dabei helfen, den richtigen Kompromiss zwischen diesen beiden Extremen zu finden.},
language = {de}
}